Citizen-Editors’ Endogenous Information Acquisition and News Accuracy∗ Francesco Sobbrio†

Abstract This appendix provides additional materials that are also discussed in the paper. Specifically, Appendix A contains the formal proofs of the propositions and lemmas presented in the paper. Appendix B discusses the robustness of the results. Appendix C presents additional empirical evidence on the negative correlation between media slant and news accuracy. Appendix D derives the probabilities of reaching a stopping threshold and presents a detailed derivation of the equations characterizing the marginal viewers. JEL Classification: D81, D83, L82 Key Words: Media Bias, Slant, Information Acquisition, Valence, Competition



A previous version of the paper circulated under the title “A Citizen-Editors Model of News Media”. I am very grateful to the editor, Brian Knight for his thoughtful and detailed editorial guidance and to two anonymous referees for their useful comments. I would also like to thank Juan Carrillo, Micael Castanheira, Francesco De Sinopoli, Matthew Ellman, Fabrizio Germano, Francois Maniquet, Andrea Mattozzi, Paolo Pin, Simon Wilkie and seminar participants at the Catholic University of Milan, Institut d’An` alisi Econ` omica-CSIC, Universitat de Barcelona, University of Bologna, Universit`e catholique de Louvain, University of Namur, University of Padova; and at SAEe 2011, Congress of the European Economic Association, 7th Workshop in Media Economics, 2009 European Meeting of the Econometrics Society, the 2009 Meeting of the Association for Public Economic Theory, the 8th Journ´ees Louis-Andr´e G´erard-Varet, the 2009 Meeting of the European Public Choice Society. The usual disclaimers apply. † Department of Economics and Finance, Catholic University of Milan, Largo Gemelli 1, 20123, Milan, Italy. Email: [email protected]

Appendix A Proof of Proposition 1 The problem involves analyzing a stochastic process with two absorbing states. Specifically, the ¯ ∗e ) must be determined.1 After equations characterizing these two absorbing states (i.e., n∗e and n m draws, given that a current difference in signals in favor of r equal to n, the value function of editor e is given by (9). First, suppose that the state of the world s = r. Then at a given point in time, for a difference in signals in favor of r equal to n, the value function of an editor with idiosyncratic preferences xe will satisfy the following second order difference equation: Ver (n) = θVer (n + 1) + (1 − θ)Ver (n − 1) − c where the associated homogenous equation is: θy 2 − y + (1 − θ) = 0 whose solutions are: y1 = 1, y2 =

1−θ θ

Moreover, since the difference equation is non-homogenous it has also a specific solution of the form Ver (n) = Hn, thus we should also find a solution of: [θH(n + 1) − H(n) + (1 − θ)H(n − 1)] = c Thus H =

c 2θ−1 .

Hence, the generic solution to the second order difference equation is: Ver (n) = a + bλn + Hn

where λ =

1−θ θ .

In order to find the values of a and b we should consider the two terminal conditions

given stopping rule n ¯ e and ne :

where µ(n) =

1 1+λn .

Ver (¯ ne ) = δ(2µ(¯ ne ) − 1) − (1 − xe )

(A-1)

Ver (ne ) = δ(1 − 2µ(ne )) − xe

(A-2)

That is (A-1) represents the utility of editor e when reaching n ¯ e signals in

favor of state r (where she chooses alternative R). Similarly (A-2) represents the utility of editor e when reaching |ne | signals in favor of state l (where she chooses alternative L). Thus, given these two terminal conditions we have that: a + bλn¯ e + H n ¯e = δ 1

1 − λn¯ e − (1 − xe ) 1 + λn¯ e

This is a standard problem of sequential testing of two simple hypotheses (see Chapter 4 in Shiryaev, 2007).

1

a + bλne + Hne = δ

λne − 1 − xe 1 + λne

Thus: Ver (n, n ¯ e , ne )

 (λn − λn¯ e ) 1 − λn¯ e − (1 − x ) + =δ e 1 + λn¯ e (λne − λn¯ e )     λne λn¯ e − 1 + H(¯ n − n 2δ ne − n) ) + 1 − 2x e e − H(¯ e (1 + λne ) (1 + λn¯ e ) 

Similarly, supposing that the state of the world s = l, we can derive Vel (n, n ¯ e , ne ) : λne (λn¯ e − λn ) 1 − λn¯ e Vel (n, n ¯ e , ne ) = δ − (1 − x ) + e 1 + λn¯ e λn (λne − λn¯ e )   n ¯ n 1 − λ eλ e 2δ ne − n) + H(¯ ne − ne ) − 1 + 2xe + H(¯ (1 + λn¯ e ) (1 + λne ) Thus the expected utility of editor e givena difference of signals in favor of r equal to n is: Ve (n, n ¯ e , ne ) =

1 λn r V ) + V l (n, n (n, n ¯ , n ¯ e , ne ) e e e 1 + λn 1 + λn e

(A-3)

Therefore, n ¯ ∗e and n∗e are the solutions of the two first order conditions: ∗   ∗ i    ∗  ∂Ve (ln λ) λn¯ e h ne ∗ n ¯ ∗e ne ∗ =0 (2x − 1) λ )) − H 1 − λ + 1 − λ − 1 (2δ − H(¯ n − n |n¯ ∗e = n∗ e e ∂n ¯e λ e − λn¯ ∗e ∗  ∗     i  ∗ ∂Ve (ln λ) λne h n ¯e ∗ ne n ¯ ∗e ∗ ∗ |n=ne = n∗ (2x − 1) λ + 1 + 1 − λ (2δ − H (¯ ne − ne )) + H λ − 1 = 0 ∂ne λ e − λn¯ ∗e

Notice that the optimal stopping rule n ¯ ∗e and n∗e do not depend on n. That is the optimal stopping rule do not change depending on the realization of the signals.2 Let’s consider the two first order conditions and let’s denote them as f and g. That is: ∂Ve |n¯ ∗ = 0 ∂n ¯ ∗e e

(A-4)

∂Ve |n=n∗e = 0 ∂n∗e

(A-5)

f=

g=

that is n ¯ ∗e and n∗e are the solution of the following system of equations: (

f (¯ n∗e (xe , δ, c), n∗e (xe , δ, c), xe , δ, c) = 0 g(¯ n∗e (xe , δ, c), n∗e (xe , δ, c), xe , δ, c) = 0

In order to obtain the comparative statics, it is necessary to derive the differential of these funcA detailed formal derivation of the second order conditions, ensuring that (¯ n∗e , n∗e ) is a global maximum, is available upon request to the author. 2

2

tions.3 That is:

  

∂f n∗e + ∂n ¯ ∗e d¯ ∂g n∗e + ∂n ¯ ∗e d¯

∂f ∗ ∂n∗e dne + ∂g ∗ ∂n∗e dne +

∂f ∂xe dxe + ∂g ∂xe dxe +

∂f ∂δ dδ ∂g ∂δ dδ

+ +

∂f ∂c dc ∂g ∂c dc

=0

dn∗e dxe

and

Let’s focus on the comparative statics with respect to xe . That is,

=0

d¯ n∗e dxe

must be determined,

holding the other parameter constants. Hence, dδ = 0 and dc = 0. Thus: dn∗e dxe

 =

∂g ∂f ∂n ¯ ∗e ∂xe



∂g ∂f ∂xe ∂ n ¯ ∗e



∂g ∂f ∂n∗e ∂ n ¯ ∗e



∂g ∂f ∂n ¯ ∗e ∂n∗e



∂g ∂f ∂n∗e ∂xe



∂f ∂g ∂n∗e ∂xe



∂g ∂f ∂n ¯ ∗e ∂n∗e



∂f ∂g ∂n ¯ ∗e ∂n∗e



similarly d¯ n∗e dxe

 =

Then, simple calculations yields:

and

 ∗ ∗ 2λne λn¯ e + 1 dn∗e =− <0 ∗ ∗ dxe H (λne − λn¯ ∗e ) (λne + 1)

(A-6)

 ∗ ∗ 2λn¯ e λne + 1 d¯ n∗e =− <0 ∗ dxe H (λn¯ ∗e + 1) (λne − λn¯ ∗e )

(A-7)

∗ ∗ ne dne Moreover, d¯ dxe > dxe if and only if:   ∗ ∗ ∗ ∗ (λne − λn¯ e ) 1 − λne λn¯ e < 0 thus since ∗



(1 − λne λn¯ e )

    > 0 for xe < 1/2 = 0 for xe = 1/2    < 0 for x > 1/2 e

(A-8)

the result follows. Let’s now focus on the comparative statics with respect to δ. Using the same methodology as the one described above: ∗ ∗ 2λne 1 − λn¯ e dn∗e =− <0 ∗ ∗ dδ H (λne + 1) (λne − λn¯ ∗e )

and

 ∗ ∗ 2λn¯ e λne − 1 d¯ n∗e = >0 ∗ dδ H (λn¯ ∗e + 1) (λne − λn¯ ∗e )

3

(A-9)

(A-10)

These comparative statics are determined by treating n as a real number. This mathematical abuse is made for technical convenience (for an analogous treatment see Brocas and Carrillo, 2009, and Brocas, Carrillo and Palfrey, 2012). At the same time, a marginal change in n ¯ ∗e and/or n∗e might have a straightforward interpretation in this context. For example, a marginal increase in the threshold required by a citizen-editor to endorse candidate j simply represents a marginal increase in the probability of such a citizen-editor requiring one more signal in favor of j to endorse her.

3

∗ ∗ ne dne Moreover, d¯ dδ > dδ if and only if:   ∗ ∗ ∗ ∗ (λn¯ e + λne ) λne λn¯ e − 1 > 0 hence given (A-8) the results follow. Finally, the comparative statics with respect to c are:  ∗ ∗ ∗ λne − 1 n∗e − n∗e ) 1 − λn¯ e λne (¯ dn∗e =− + >0 ∗ ∗ ∗ dc c (ln λ) (λne + 1) c (λne − λn¯ ∗e ) (λne + 1)

(A-11)

 ∗ ∗ ∗ λn¯ e (¯ n∗e − n∗e ) λne − 1 1 − λn¯ e d¯ n∗e − n¯ ∗ = <0 ∗ dc c (ln λ) (λn¯ ∗e + 1) (λ e + 1) (λne − λn¯ ∗e ) c

(A-12)

and

Q.E.D. Proof of Proposition 2 I first prove how the expected number of signals collected by an editor vary with the editor’s ideological preferences and with the parameters of the model. Then, I turn to the analysis of the expected probability of endorsing the low-valence candidate. The expected number of signals collected by an editor before endorsing a candidate is: ¯ ∗e ) · n ¯ ∗e E(|ne |) = Pr(ne = n∗e ) · |n∗e | + Pr(ne = n that is:



E(|ne |) = Clearly,

n ¯ ∗e λne − 1

  ∗ ∗ ∗ λn¯ e + 1 − n∗e λne + 1 1 − λn¯ e ∗ 2 (λne − λn¯ ∗e )

∂E(|ne |) d¯ n∗e ∂E(|ne |) dn∗e dE(|ne |) = + dxe ∂n ¯ ∗e dxe ∂n∗e dxe

where



n∗ e

λ

 −1

∂E(|ne |)  =  ∗ n ∂n ¯ ∗e 2 λ e − λn¯ ∗e and



(ln λ) λ

n ¯ ∗e

λ

n∗ e

n∗ e

λ

+1



(n∗e ∗ n ¯ λ e

+

n ¯ ∗e )



n ¯ ∗e

+ λ

+1



!

 ∗   n∗  ∗ 1 − λn¯ e ∂E(|ne |) ¯ ∗e  e n∗e ne + n n ¯ ∗e = (ln λ) λ 1+λ − λ +1 ∗ n∗ ∂n∗e 2 (λne − λn¯ ∗e ) λ e − λn¯ ∗e

thus given (A-6) and (A-7) derived in the proof of Proposition 1, the sign of

dE(|ne |) dxe

will be

equivalent to the one of the following expression: n ¯ ∗e n∗e



n∗e

Exe = (1−λ λ ) λ

!    ∗ ∗ ∗ ∗ ∗ ∗ λ2¯ne λne + 1 λ2ne − 1 λ2ne λn¯ e + 1 1 − λ2¯ne + ∗ (λn¯ ∗e + 1) (λne + 1)

 (ln λ) (n∗ + n ¯ ∗e ) e +λ − ∗ λne − λn¯ ∗e n ¯ ∗e

Thus, given (A-8) and given that:

(n∗e + n ¯ ∗e )

    > 0 for xe < 1/2 = 0 for xe = 1/2    < 0 for x > 1/2 e

4

(A-13)

then:

    > 0 for xe < 1/2

dE(|ne |) = 0 for xe = 1/2 dxe    < 0 for x > 1/2 e Let’s now focus on δ. Since dE(|ne |) ∂E(|ne |) d¯ n∗e ∂E(|ne |) dn∗e = + ∗ dδ ∂n ¯e dδ ∂n∗e dδ and given (A-9) and (A-10) derived in the proof of Proposition 1, the sign of

dE(|ne |) dδ

will be

equivalent to the one of the following expression:  2  ∗ ∗ ∗ ∗ ∗ ∗ 2 λne λne + 1 1 − λn¯ e λn¯ e λn¯ e + 1 λne − 1 + Eδ = ∗ (λn¯ ∗e + 1) (λne + 1) ∗ ∗ ∗ ∗ ∗ ∗ ¯ ∗e ) (1 − λne λn¯ e )(1 + λne λn¯ e ) λ2ne − λ2¯ne (ln λ) (n∗e + n − ∗ ∗ (λne − λn¯ ∗e ) (λne + 1) (λn¯ ∗e + 1) ∗



where the above expression is always positive since (n∗e + n ¯ ∗e ) and (1 − λne λn¯ e ) are both positive for xe < 1/2 and both negative for xe > 1/2. Moreover, for xe = 1/2 the above expression reduces 2  ) ( n¯ ∗ | 2 1−λ e xe =1/2 to > 0. ( n¯ ∗ | ) λ e xe =1/2 Let’s now focus on c. Since ∂E(|ne |) d¯ n∗e ∂E(|ne |) dn∗e dE(|ne |) = + dc ∂c dc ∂n∗e dc and given (A-11) and (A-12) derived in the proof of Proposition 1, the sign of

dE(|ne |) dc

will be

equivalent to the one of the following expression:     ∗ − n∗ ) λn∗e (1 − λn ¯ ∗e ) + λn ¯ ∗e λn∗e − 1 2 ∗ n ¯ (¯ n e e e −1 2−λ Ec = − ∗ c ln λ c (λne − λn¯ ∗e )  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (n∗e + n ¯ ∗e ) (λne λn¯ e − 1) λne − 1 1 − λn¯ e (ln λ) (n∗e + n ¯ ∗e ) (λ2ne λ2¯ne − 1) (¯ n∗e − n∗e ) λ2ne − λ2¯ne + − ∗ ∗ ∗ c (λne + 1) (λn¯ ∗e + 1) c (λne − λn¯ ∗e )2 (λne + 1) (λn¯ ∗e + 1) ∗ λne

where the above is always negative. Indeed the first two terms are clearly negative and the second ∗



and third negative are always negative since (n∗e + n ¯ ∗e ) and (λne λn¯ e − 1) have always opposite signs ∗



for xe < 1/2 or xe > 1/2 (similarly, (n∗e + n ¯ ∗e ) and (λ2ne λ2¯ne − 1) have also always opposite signs for xe < 1/2 or xe > 1/2). Moreover, for xe = 1/2 the above expression reduces to: 

∗ 2 − λ( n¯ e |xe =1/2 )

h   i ∗ ∗ (1 − λ2( n¯ e |xe =1/2 ) ) − 2 (ln λ) n ¯ ∗e |xe =1/2 λ( n¯ e |xe =1/2 )   <0 ∗ ∗ c (ln λ) λ( n¯ e |xe =1/2 ) 1 + λ( n¯ e |xe =1/2 )

This proves the first part of the Proposition relative to the comparative statics on the expected number of signals collected by an editor before endorsing a candidate. Let’s now analyze the

5

probability of an editor endorsing the low-quality candidate. Since: Pr(τe = L|s = r) = and Pr(τe = R|s = l) =

2µ(¯ n∗e ) − 1 µ(n∗e ) µ(¯ n∗e ) − µ(n∗e )

1 − 2µ(n∗e ) [1 − µ(¯ n∗e )] µ(¯ n∗e ) − µ(n∗e )

Thus it is easy to verify that Pr(τe = L|s = r) is decreasing in xe and Pr(τe = R|s = l) is increasing in xe . Moreover, the ex-ante probability of making a wrong choice is: Pr(error) = Pr(s = r) Pr(τe = L|s = r) + Pr(s = l) Pr(τe = R|s = l) hence:4





Pr(error) =



λn¯ e (λne − 1) + (1 − λn¯ e ) ∗ 2 (λne − λn¯ ∗e )

It is now possible to perform the comparative statics upon this probability. First of all: 2 ∗ λne − 1 ∂ Pr(error) 1 n ¯ ∗e = (ln λ) λ <0 ∗ ∂n ¯ ∗e 2 (λne − λn¯ ∗e )2 ∗ 2 1 − λn¯ e ∂ Pr(error) 1 n∗e = − (ln λ) λ >0 ∗ ∂n∗e 2 (λne − λn¯ ∗e )2

Hence, since d Pr(error) dδ

d¯ n∗e dc

< 0 and

dn∗e dc

> 0, then

d Pr(error) dc

> 0. Similarly, since

d¯ n∗e dδ

> 0 and

dn∗e dδ

< 0, then

< 0.

Finally given (A-6) and (A-7) derived in the proof of Proposition 1,

d Pr(error) dxe

> 0 if and only if:

∗ ∗ ∗ ∗ ∗ ∗ λ2ne − λ2¯ne 1 + λne λn¯ e 1 − λne λn¯ e <0 ∗ (λne + 1) (λn¯ ∗e + 1)

Thus, given (A-8):     < 0 for x <

d Pr(error) = 0 for x =  dxe   > 0 for x >

1 2 1 2 1 2

Q.E.D.

4 n∗ e

λ

Accordingly, the probability of endorsing the high valence candidate is 1 − Pr(error) ∗



¯e (1−λn )+(λne −1) ∗ ¯∗ e) 2(λne −λn

6

=

Proof of Lemma 1 It is immediate to verify that (˜ xe − x ˆe ) is decreasing in C. Let’s now focus on x ˜e . Then:  ∗ ∗ λn¯ e λne − 1 d˜ xe (¯ n∗e , n∗e ) >0 = −C (ln λ) ∗ d¯ n∗e (λne + 1) (1 − λn¯ ∗e )2 ∗

d˜ xe (¯ n∗e , n∗e ) λne = (ln λ) ∗ dn∗e (λne + 1)2

! ∗ λn¯ e + 1 <0 2δ − C (1 − λn¯ ∗e )

The it is immediate to verify that x ˜e is increasing in δ. Let’s now analyze how x ˜e changes as xe increases. First, I want to prove that for any xe < 1/2 it is always xe /dxe > 0. From the case that d˜ dn∗e d¯n∗e the proof of Proposition 1 we know that for xe < 1/2, dxe > dxe . Hence, a sufficient condition to ensure that d˜ xe /dxe > 0 is simply: d˜ n∗e , n∗e ) d˜ xe (¯ n∗e , n∗e ) xe (¯ > dn∗e d¯ n∗e which is true if and only if: C

Since

∂ ∂n ¯ ∗e





¯e 1−λn ∗ n 1+λ ¯ e



> 0 and

d¯ n∗e dxe

 ∗ ∗ λn¯ e λ2ne − 1

! ∗ λn¯ e + 1 < 2δ + ∗ (1 − λn¯ ∗e ) λne (1 − λn¯ ∗e )2

< 0, then δ





¯e 1−λn ∗ n 1+λ ¯ e



≥ C max . Hence, a sufficient condition for the

above condition to be always true is:   ∗ ∗ ∗ ∗ (λn¯ e λne − 1) λn¯ e + λne < 0 which it is always the case for xe < 1/2. Moreover, for xe = 1/2, n∗e = −¯ n∗e and thus:  ∗ ∗ ∗ λ2¯ne δ(1 − λn¯ e ) − C(λn¯ e + 1) d˜ xe 4 = − (ln λ) >0 dxe xe =1/2 H (1 − λ2¯n∗e ) ((1 − λ3¯n∗e ) + λn¯ ∗e (1 − λn¯ ∗e )) Hence, for any xe ≤ 1/2, it is always the case that d˜ xe /dxe > 0. Let’s analyze now the case where xe > 1/2. Then, d˜ xe /dxe > 0 if and only if: ∗

C < C˜ ≡ 2δ



λ2ne 1 − λ2¯ne

2

λ2¯n∗e (λ2ne − 1) (λne + 1)2 + λ2ne (λn¯ ∗e + 1)2 (1 − λ2¯n∗e ) ∗





(A-14)

hence C˜ > 0. Let’s now analyze how C˜ changes when xe increases: ∗ ∗ ∗ 4δ (ln λ) 1 − λ2¯ne λ2ne +¯ne ∂ C˜ = − Y >0 ∗ ∗ ∗ ∂n ¯ ∗e λ2¯n∗e (λne + 1)2 (λ2ne − 1) + λ2ne (λn¯ ∗e + 1)2 (1 − λ2¯n∗e )

where ∗

 ∗

Y = 2λn¯ e +

1 − λ2¯ne

 h 2n∗ n¯ ∗  2 2 2n∗ i  ∗ ∗ ∗ ∗ ∗ ∗ λ e λ e + 1 1 − λ2¯ne − λn¯ e + 1 λ2ne +¯ne + λn¯ e λne + 1 λ e −1 >0   ∗ ∗ ∗ ∗ ∗ ∗ 2 2 n 2n 2n 2¯ n n ¯ 2¯ n e e e e e e λ (λ + 1) (λ − 1) + λ (λ + 1) (1 − λ )

7





since λn¯ e λne + 1

2

 2 ∗ ∗ ∗ ∗ λ2ne − 1 > λn¯ e + 1 λ2ne +¯ne > 0. Moreover:

2 n¯ ∗ 2 n ∗  ∗ ∗ ∗ ∗ ∗ 4δ (ln λ) λ2ne λn¯ e − 1 λ e +1 λ e + λ4ne + λ3ne + 1 λ2¯ne ∂ C˜ >0 =− 2  ∂n∗e ∗ ∗ ∗ ∗ ∗ ∗ 2 2 n 2n 2n 2¯ n 2¯ n n ¯ e e e e e e λ (λ + 1) (λ − 1) + λ (λ + 1) (1 − λ ) hence since

d¯ n∗e dxe

< 0 and

dn∗e dxe

<0: dC˜ ∂ C˜ d¯ ∂ C˜ dn∗e n∗e = + <0 dxe ∂n ¯ ∗e dxe ∂n∗e dxe

˜ That is, since Hence, x ˜e will be increasing in xe for xe > 1/2 if and only if C < C. be increasing in xe only as long as xe <

xmax eR ,

˜ dC dxe

< 0, x ˜e will

where:

 ∗ max C˜ n ¯ ∗e (xmax eR ), ne (xeR ) = C Moreover, since

˜ dC dxe

 < 0, C˜ max < lim C˜ = C max . Finally, since δ ∈ 0, 12 , xmax eR < 1. Specifically, xe →1/2

for δ < 1/2 an editor with preferences xeR = 1 would never endorse a leftist candidate since, ˜ e = 1) = 0. For δ = 1/2, an trivially, µ ˆ(xe =1) = 0 (i.e., n ¯ ∗e (xeR = 1) = 0) which implies that C(x R R editor with preferences xeR = 1 will endorse a leftist candidate if and only if µ(n) = 0. That is, if and only if n = −∞. Hence, necessary conditions for this to be verified are n e (xe R = 1) → −∞ n∗e dn∗e and n ¯ e (xeR = 1) → 0. As shown by Proposition 1, for xe > 12 it is the case that d¯ dxe > dxe . That ∗ ∗ ∗ is, when xe → 1 it must be the case that n ¯ e → 0 but n ¯ e − ne 9 −∞. In turn, this implies R

R

that C˜ → 0 when xeR → 1.

Finally, since

˜ dC dδ

R

R

> 0, if δ increases, C˜ increases as well, thus xmax eR

will also be higher. Let’s now focus on x ˆe . Then:  ∗ ∗ ∗ λn¯ e 2δ(λne − 1) − C(λne + 1) dˆ xe = (ln λ) <0 ∗ d¯ n∗e (λne − 1) (λn¯ ∗e + 1)2 ∗ dˆ xe 1 − λn¯ e n∗e = −C (ln λ) λ >0 ∗ dn∗e (λn¯ ∗e + 1) (λne − 1)2 Thus it is immediate to verify that x ˆe is decreasing in δ. Let’s now analyze how x ˆe changes as xe increases. First, I want to prove that for any xe > 1/2 it is always xe /dxe > 0. As the case that dˆ dn∗e d¯n∗e shown in the proof of Proposition 1, for xe > 1/2, then dxe < dxe . Hence, a sufficient condition to ensure that dˆ xe /dxe > 0 is simply: ∗ , n∗ ) ∗ , n∗ ) dˆ dˆ x (¯ n x (¯ n e e e e e e < dn∗e d¯ n∗e that is



C moreover since C max ≤



λn¯ ∗e (λne



¯e 1−λn δ (λn¯ ∗e +1)



λne 1 − λ2¯ne





(λne + 1) + ∗ (λne − 1) − 1)2

! < 2δ



< δ

λne −1 , (λn∗e +1)

a sufficient condition for the above to be verified

8

becomes:

∗ ∗ ∗ ∗ (1 − λn¯ e λne ) λn¯ e + λne <0 ∗ λn¯ ∗e (λ2ne − 1) ∗



hence since (1 − λn¯ e λne ) < 0 for xe > 1/2, we have proved that for xe > 1/2 it is always the case n∗e and thus: that dˆ xe /dxe > 0. Moreover, for xe = 1/2, n∗e = −¯  ∗ ∗ ∗ λ2¯ne δ(1 − λn¯ e ) − C(λn¯ e + 1) dˆ xe 4 >0 = − (ln λ) dxe xe =1/2 H (1 − λ2¯n∗e ) ((1 − λ3¯n∗e ) + λn¯ ∗e (1 − λn¯ ∗e )) Hence, for any xe ≥ 1/2, it is always the case that dˆ xe /dxe > 0. Let’s now analyze the case where xe < 1/2. In this case, dˆ xe /dxe > 0 if and only if: C < Cˆ ≡ 2δ

2 ∗ ∗ λ2¯ne λ2ne − 1

(A-15)

λ2ne (1 − λ2¯n∗e ) (λn¯ ∗e + 1)2 + λ2¯n∗e (λ2ne − 1) (λne + 1)2 ∗





hence Cˆ > 0. Let’s now analyze how Cˆ changes when xe increases. First of all: ∂ Cˆ ∗ = 4δ (ln λ) λ2¯ne  ∂n ¯ ∗e and

∂ Cˆ = ∂n∗e

2 n¯ ∗  ∗ ∗ ∗ ∗ λ2ne − 1 λ e + λ4¯ne + λ3¯ne + 1 λ2ne λ2¯n∗e

∗ (λne

+ 1)

2

∗ (λ2ne

− 1) +

∗ λ2ne

(λn¯ ∗e

2

+ 1) (1 −

λ2¯n∗e )

2 < 0

 ∗ ∗ ∗ 4δ (ln λ) λ2ne − 1 λne +2¯ne

2 W < 0 ∗ ∗ ∗ λ2¯n∗e (λne + 1)2 (λ2ne − 1) + λ2ne (λn¯ ∗e + 1)2 (1 − λ2¯n∗e )

where      ∗  ∗  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ W = 2λn¯ e λne (λ2ne − λ2¯ne ) + 1 + λ2ne λn¯ e 1 − λn¯ e + λne − λ2¯ne λ2ne + λne λ2¯ne + λ3ne λ2¯ne + 1 hence since

d¯ n∗e dxe

< 0 and

dn∗e dxe

<0: dCˆ ∂ Cˆ d¯ n∗e ∂ Cˆ dn∗e = + >0 dxe ∂n ¯ ∗e dxe ∂n∗e dxe

ˆ That is, since Hence, x ˆe will be increasing in xe for xe < 1/2 if and only if C < C.

ˆ dC dxe

> 0, x ˜e will

min be increasing in xe only as long as xe > xmin eL , where xeL is such that:

 ∗ min Cˆ n ¯ ∗e (xmin eL ), ne (xeL ) = C ˆ dC dxe

> 0, Cˆ max < lim Cˆ = C max . Finally, by using an analogous proof to the one xe →1/2  1 employed above to show that xmax eR < 1, it is immediate to see that since δ ∈ 0, 2 , it is always ˆ the case that xmin > 0 and that Cˆ → 0 when xe → 0. Finally, since dC > 0, if δ increases, Cˆ e Moreover, since

L

L

increases as well, thus

xmin eL

will be smaller.



Q.E.D.

Proof of Proposition 3 The optimal strategy for a profit maximizing monopolist media outlet is to choose an editor with

9

idiosyncratic preference xe such that its profits are maximized. That is xmon must be such that: e dΠ dΠ d¯ dΠ dn∗e n∗e = + =0 dxe d¯ n∗e dxe dn∗e dxe Where: dΠ dF (˜ xe ) dF (ˆ xe ) = − d¯ n∗e d¯ n∗e d¯ n∗e dΠ dF (˜ xe ) dF (ˆ xe ) = − dn∗e dn∗e dn∗e where

dF (˜ xe ) d¯ n∗e

=

d d¯ n∗e

Z

x ˜e (¯ n∗e )

f (x)dx. Hence applying Leibniz’s rule: δ

dF (˜ xe ) d = d¯ n∗e d¯ n∗e thus,

x ˜e (¯ n∗e ,n∗e )

Z

f (x)dx = f (˜ xe (¯ n∗e , n∗e ))

δ

d˜ xe (¯ n∗e , n∗e ) d¯ n∗e

dΠ xe (¯ n∗e , n∗e ) xe (¯ n∗e ) ∗ ∗ d˜ ∗ ∗ dˆ = f (˜ x (¯ n , n )) − f (ˆ x (¯ n , n )) e e e e e e d¯ n∗e d¯ n∗e d¯ n∗e

similarly

dΠ xe (¯ n∗e , n∗e ) xe (¯ n∗e , n∗e ) ∗ ∗ d˜ ∗ ∗ dˆ = f (˜ x (¯ n , n − f (ˆ x (¯ n , n )) )) e e e e e e dn∗e dn∗e dn∗e

Hence the first order condition becomes: d˜ xe /dxe f (ˆ xe (¯ n∗e , n∗e )) = dˆ xe /dxe f (˜ xe (¯ n∗e , n∗e ))

(A-16)

where: −2 (ln λ) d˜ xe = ∗ dxe H(λne − λn¯ ∗e ) dˆ xe −2 (ln λ) = ∗ dxe H(λne − λn¯ ∗e )







λ2ne λn¯ e + 1 ∗





−C

(λne + 1)3 ∗

λ2¯ne λne + 1 (λn¯ ∗e + 1)3





λ2¯ne λne − 1



(λn¯ ∗e + 1) (1 − λn¯ ∗e )2 ∗

−C



λ2ne 1 − λn¯ e





+

+



λ2ne λn¯ e + 1

2

!!

(1 − λn¯ ∗e ) (λne + 1)3 ∗





(λne + 1) (λne − 1)2 ∗







λ2¯ne λne + 1

2

!!

(λne − 1) (λn¯ ∗e + 1)3 ∗

From the proof of Lemma 1, we know that for xe = 1/2, d˜ xe /dxe = dˆ xe /dxe > 0. Hence, for xe = 1/2,

d˜ xe /dxe dˆ xe /dxe

= 1. More generally, for any xe : xe d˜ xe dˆ ∗ ∗ ∗ ∗ − = (1 − λne λn¯ e )(λne − λn¯ e ) · α · β dxe dxe

where α = 2δ and

∗ ∗ ∗ ∗ ∗  ∗ 4λn¯ e λne + λne + λn¯ e (1 + λne λn¯ e )

(λne + 1)3 (λn¯ ∗e + 1)3 ∗

 β = 4C





ne λ2¯ λ2ne 2 2 ∗ ∗ + ∗ n ¯ 2¯ n n e e e (λ +1) (1−λ ) (λ +1) (λ2n∗e −1) ∗ (λne − 1) (1 − λn¯ ∗e )

10



where α and β are always positive. Hence given (A-8):     > 1 for xe <

d˜ xe /dxe = 1 for xe = dˆ xe /dxe    < 1 for x > e In other words, for xe >

1 2

1 2 1 2 1 2

(A-17)

an increase in xe increases x ˆe more than x ˜e (and viceversa for xe < 12 ).

Then, it is immediate to verify that when the distribution of citizens’ idiosyncratic preferences is such that Condition A is verified, then xe =

1 2

is the unique stationary point and the global

maximum. Now suppose F (x) is such that Condition B is verified. For xeR >

1 2

to be a stationary point it

xeR (¯ n∗e , n∗e )). Moreover, from Lemma 1 and (A-17) we must be the case that f (ˆ xeR (¯ n∗e , n∗e )) < f (˜ ˆe R (¯ n∗e , n∗e ). Then, xe = 12 cannot be a global know that for xeR > 1/2, then x ˜eR (¯ n∗e , n ∗e ) > 1 − x (x) xe dˆ xe 1 mon > 0 and d˜ maximum since dfdx dxe x=1/2 = dxe x=1/2 . Thus the stationary point xeR > 2 x=1/2  such that (A-16) is satisfied will be a global maximum on 12 , 1 . Then by the symmetry of f, choosing an editor with symmetric preferences will also be profit-maximizing. That is, we have mon mon two global maxima in this case xmon eR and xeL = 1 − xeR . Indeed, since the distribution function

f is symmetric around

1 2,

so it must be the demand function. To sum up, if F (x) is such that

Condition A holds the global maximum is always at xe = 21 . Instead, if F (x) is such that Condition B holds, there are two symmetric global maxima such that xeR = 1 − xeL > 1/2. The last part of the proposition follows immediately from Lemma 1

Q.E.D.

Proof of Proposition 4 Let’s start with the case where Condition A holds. We show that in this case the unique equilibrium is such that x1e = x2e = 21 . Suppose that media outlet 1 deviates by choosing x1e > x2e = 12 . If media outlet one deviates, the indifferent viewer, i.e., the viewer who will be indifferent between watching media outlet 1 and media outlet 2 is the one having preferences xI such that UI (W1 ) = UI (W2 ). That is:   ∗  ∗ ∗  ∗   (1 − λn¯ e2 ) λne1 − λn¯ e1 ∗   ∗  − λne1 − 1 1 − λn¯ e1  n ¯ e2 −1 λ +1 

xI (¯ n∗e1 , n∗e1 , n ¯ ∗e2 ) =

δ 1 + ∗ ∗ 2 λne1 λn¯ e1

where since x2e = 21 , then n ¯ ∗e2 = −n∗e2 . The no-deviation condition for media outlet 1 requires that @xe >

1 2

such that the demand if deviating is higher than the demand if not deviating. Specifically,

the demand that media outlet 1 faces when not deviating is: D

N Dev

(x1e )

=D

N Dev

x2e



  i  1 1h F(x ˜e |xe = 1 ) − F ( x ˆ|xe = 1 ) = F ( x ˜e |xe = 1 ) − F = 2 2 2 2 2

(A-18)

Instead the demand that media outlet 1 faces if it deviates is: h i DDev (x1e ) = F ( x ˜e |x1e ) − F (max {ˆ xe1 ; xI (xe1 )})

11

(A-19)

Notice that for any non-uniform distribution satisfying Condition A the mass of citizens is strictly decreasing moving away from the mean of the distribution at 1/2. Hence it is enough to show that this no-deviation condition holds even in the case where citizens’ preferences are uniformly distributed in [0, 1].5 In the case of a uniform distribution, the following represents a sufficient no-deviation condition: ¯ ∗e2 ) − xI (¯ n∗e1 , n∗e1 , n

1 >x ˜e |x1e − x ˜e |xe = 1 2 2

hence media outlet 1 would not deviate if and only if:    ∗   ∗ ∗ ∗ n ¯ ∗e2   λne1 − λn¯ e1 λ2ne1 − 1 1 − λn¯ e1 ∗ (1 − λ )    − ∗  λn¯ e1 + 1  =δ 2  ∗ ∗ ∗ n n ¯ λ e1 + 1 λ e2 + 1 λne1 λn¯ e1 − 1 

C > C T HR

where C T HR > 0 if and only if  ∗ ∗ ∗ λne1 − λn¯ e1 (1 − λn¯ e2 ) > ∗  ∗   ∗ λne1 + 1 λn¯ e1 + 1 λn¯ e2 + 1 

Let A =

  n ¯∗ e n∗ λ e1 −λ 1 .  ∗  n ¯∗ e n λ e1 +1 λ 1 +1

For xe > 12 ,

dA dxe

< 0 which implies that:

  ∗  ∗ ∗ ∗ λne1 − λn¯ e1 λne1 − λn¯ e1  ∗  ∗ <  ∗  ∗  λne1 + 1 λn¯ e1 + 1 λne1 + 1 λn¯ e1 + 1 



xe = 12

(1 − λn¯ e2 )  = ∗ λn¯ e2 + 1

(A-20)

hence C T HR < 0. Therefore, in a duopoly when the distribution of citizens’ idiosyncratic preferences is such that Condition A holds (and where citizens watch at most one media report), there will never be an incentive to deviate from the equilibrium at x1e = 1 − x2e = 21 . Moreover, notice that this is the unique Nash equilibrium. If the two media outlets were to choose ideological editors, then each of them would clearly have an incentive to deviate by choosing a moderate one. Let’s now analyze the case where Condition B holds. First of all, in order to ensure that there is someone willing to watch media 1 the following condition must be satisfied ¯ ∗e2 ) < x ˜e (x1e ) xI (¯ n∗e1 , n∗e1 , n that is:

 C < C¯ = 2δ 



1 − λn¯ e1 ∗

λn¯ e2 + 1

 (A-21)



¯ Consider (A-18) where clearly C¯ > 0.6 Let’s now analyze the no-deviation condition for C < C. 5

Notice also that, as stated in section 2.1 the analysis focuses on symmetric distributions.  

˜e |xe = 1 if and only if C < Cˇ ≡ 2δ Notice also that xI (¯ n∗e1 , n∗e1 , n ¯ ∗e2 ) < x 2 ∗ ¯ ∗e . since ne > n 6

1

2

12

n ¯∗ e1

1−λ

n ¯∗ λ e2 +1

n∗

n ¯∗

λ e1 λ e2 −1 n∗ n ¯∗ λ e1 λ e1 −1

where Cˇ > 0

and (A-19) and let C Duop = C Duop (xe1 ) be the highest opportunity cost such that for xe1 ∈

1 2, 1



the following condition holds (i.e., C Duop being the opportunity cost associated with the most profitable deviation from xe1 = 1/2):7 F (max {ˆ xe1 ; xI (xe1 )}) − ˜e |xe = 1 ) F (˜ xe1 ) − F ( x

1 2

≥0

(A-22)

2

  ¯ C Duop , then for C ∈ 0, C Dev media outlet 1 will have an incentive now denote C Dev = min C, to deviate by choosing an ideological editor.8 Hence, in such case there is no equilibrium where both media outlets choose a moderate editor.9 Let’s now show that it can never exist an equilibrium with xe1 = xe2 6= (e.g., xe1 = xe2 >

1 2. 1 2 ).

Suppose the two media outlets choose the same type of ideological editors By doing so their demand would be D1 (xe1 = xe2 ) = D2 (xe1 = xe2 ) =

xe1 ) F (˜ xe1 ) − F (ˆ 2

while if media outlet 2 chooses an editor with preferences xe2 = 1 − xe1 its demand would be: 2

D (xe2



1 = 1 − xe1 ) = min F (˜ xe2 ); 2

 − F (ˆ xe2 )

where by symmetry x ˆe2 = 1 − x ˜e1 , which implies that F (ˆ xe2 ) = 1 − F (˜ xe1 ) . Thus, a necessary condition for media outlet 2 not be willing to deviate is 1 − F (ˆ xe1 ) > F (˜ xe1 ) . However, since 1 2,

then x ˜ e1 > 1 − x ˆe1 and given Condition B this condition cannot hold. An analogous  proof applies for xe1 = xe2 < 12 . Hence, for C ∈ 0, C Dev the only possible Nash Equilibrium must

xe1 >

be such that xe1 = 1 − xe2 6= 1/2. Let’s show that this is indeed an equilibrium.10 Suppose that  xe1 = 1 − xe2 > 12 , then there are two possible cases. In the first one, ∀xe1 = 1 − xe2 ∈ 12 , xmax eR it is always the case that:11 dF (max {ˆ xe1 ; xI (xe1 )}) dxe1 xe

1 =1−xe2

dF (˜ xe1 ) < dxe1 xe

(A-23) 1 =1−xe2

where for xe1 = 1 − xe2 , xI (xe1 ) is always 1/2. Hence in this case xe1 = 1 − xe2 = xmax eR is a Nash equilibrium. Indeed, by Lemma 1, x ˜e1 is increasing in xe1 if and only if xe1 < xmax eR . Hence, dF (˜ xe 1 ) max > 0 if and only if xe1 < xeR . On the other hand, since by Lemma 1, for dxe 1

xe1 =1−xe2

xe1 > 1/2, x ˆe1 is always increasing in xe1 and xI is increasing in xe1 when xe1 = 1 − xe2 (i.e., for dF (max{x ˆe1 ;xI (xe1 )}) xI = 1/2). Thus given Condition B it is always the case that > 0. dxe 1

7

xe1 =1−xe2

Since f (x) is assumed to be symmetric with respect to 1/2, the mean and the median will always be at 1/2. Hence F (1/2) = 1/2. 8 Clearly, if C Dev < 0, firm 1 will never have an incentive to deviate. Indeed, as shown in the previous case where (Condition A) holds, when F is a uniform c.d.f. C Dev = C T HR < 0. 9 Dev C is always lower than C max since for C = C max only citizens with xe = 21 watch news reports and thus firm 1 would never have anincentive to deviate. 10 Obviously, for C ∈ 0, C Dev there are always two symmetric Nash Equilibria, i.e., xe1 = 1 − xe2 < 21 and xe1 = 1 − xe2 > 12 . 11 Symmetric conditions apply for media outlet 2.

13

Thus, none of the two media outlet would have an incentive to deviate from xe1 = 1 − xe2 = xmax eR  1 max by choosing a more leftist or more rightist editor. In the second case, ∃xe1 ∈ 2 , xeR such that (A-23) is not verified. Hence, since by construction of C Dev , for C < C Dev : dF (max {ˆ xe1 ; xI (xe1 )}) dxe1 xe 1 max 2 , xeR

then it will always exist a xe1 ∈



1 =1−xe2 =1/2

1 max 2 , xeR



1 =1−xe2 =1/2.

such that:

dF (max {ˆ xe1 ; xI (xe1 )}) dxe1 xe that is, xe1 = 1 − xe2 ∈

dF (˜ xe1 ) < dxe1 xe

1 =1−xe2

dF (˜ xe1 ) = dxe1 xe

(A-24) 1 =1−xe2

is a Nash equilibrium. Finally, we need to show that a lower C

is associated with a Nash equilibrium where the difference between the idiosyncratic preferences of the editors chosen by each media outlet, i.e., |xe1 − xe2 | , is higher. First of all by Lemma 1, a d˜ xe (¯ n∗e ,n∗ )

min 1 e1 lower C corresponds to a higher xmax eR and a lower xeL . Moreover, since as C decreases dxe dF (˜ x ) increases, hence dxee1 increases as well. Thus since the RHS of (A-24), the LHS must 1

xe1 =1−xe2

increase as well, which, in turn implies that xe1 must be higher (similarly, xe2 will be lower). That is, a lower C is associated with an equilibrium where the two media outlets choose less moderate editors.

Q.E.D.

Proof of Proposition 5 We have to analyze the no-deviation condition with K media outlets. Let n ¯ ∗e = −n∗e be the stopping thresholds chosen by a moderate editor. The demand media outlet 1 faces if it chooses a moderate editor as all the other media outlets is ∀j ∈ {2, 3, ....., K}: D

N Dev

(x1e )

=D

N Dev

xje



  i 1 h 2 1 = F(x ˜ | xe = 1 ) − F ( x ˆ|xe = 1 ) = F(x ˜ | xe = 1 ) − F ( ) 2 2 2 K K 2

Instead the demand that media outlet 1 faces if it deviates from such position is: D

Dev

(x1e )

h i = F(x ˜|x1e ) − F (max {ˆ xe1 ; xI (xe1 )})

Hence given a uniform distribution, media outlet 1 will prefer not to choose a moderate editor if and only if: K > K∗ =

i h 2 x ˜|xe = 1 − 12 2

x ˜|x1e − max {ˆ xe1 ; xI (xe1 )}

where we know from the proof of Proposition 4, that K ∗ > 2. Moreover, the game satisfies the properties of Theorem 4 in Dasgupta and Maskin (1986) for the existence of an equilibrium in a product competition game. Hence, the K ∗ media outlets game possesses a symmetric mixed∗

strategy Nash equilibrium. Moreover it is always the case that dK ˆe1 is increasing in C dC > 0 since x and d˜ xe /dxe is decreasing in C.

Q.E.D.

14

Appendix B B.1

Media owners and citizen-editors

Since the main focus of the paper is on the demand for slanted news, the model provides a stylized representation of media outlets’ profits. Considering a more general compensation mechanism for the editor would affect both the revenues and the costs of a media outlet. Once on the job, editors (and journalists) are the ones who will spend time and exert effort to collect evidence on any given issue. That is, media outlets/media owners do not directly bear this day to day cost of information acquisition. Nevertheless, in order to increase its profits, a media outlet may try to induce its editor to change her optimal information acquisition strategy by designing an incentive mechanism. As shown by Lemma 1, ideally all citizens would like to watch a media outlet whose editor keeps acquiring information until she learns the true state of the world (i.e., ¯ ∗e = ∞). However, it is not feasible for the media outlet to induce the editor to adopt n∗e = −∞, n such a sampling strategy. This is true for two simple reasons: i) information acquisition is costly for the editor and hence it is also costly for the media outlet to compensate the editor for acquiring extra pieces of information; ii) the media outlet cannot monitor the information gathered by the editor (i.e., the media outlet cannot observe the draws sampled by the editor). Nevertheless, a media outlet may induce an editor to choose stopping rules which are higher (in absolute value) with respect to the ones she would choose in the absence of any incentive mechanism. In this perspective, a simple incentive mechanism that the media outlet could implement is to offer to the editor a share α of the media outlet’s profits. This would induce the editor to choose higher (in absolute value) stopping rules. Indeed, in the absence of perfect monitoring, an incentive scheme rewarding the editor for each extra piece of evidence collected would produce the same results of a decrease in the marginal cost of sampling c (i.e., any signal acquired is more valuable or, equivalently, less costly). That is, as shown by Proposition 1, a lower c induces an editor to acquire more information.12 Similarly, the media outlet (or, more generally, the market for news) may provide an editor with a “reputation premium” when her news reports turn out to be accurate (i.e., when endorsing the high-valence candidate). That is, the editor may receive an extra positive payoff when her choice over candidates match the true state of the world. It is immediate to see how such an incentive mechanism is equivalent to increasing the value of the valence parameter δ in the editor’s utility function. Hence, as shown by Proposition 1, the presence of a “reputation premium” would induce editors to acquire more information before producing a news report. Therefore, incentive mechanisms aimed at decreasing the (net) marginal cost of sampling or at increasing the editor’s valence parameter would, indeed, increase the informativeness of the editor’s news reports. Nevertheless, such incentive mechanisms would not change the main results of the 12 Notice that a media outlet may also decrease c by giving the editor more resources to produce the news reports (e.g., more correspondents, better technology, more resources to investigate an issue, etc.).

15

model since the stopping rules of ideological editors would still be asymmetric. Indeed, as shown by Proposition 1, the presence of a private value component in the editor’s utility function always results in an ideological editor adopting a slanted information acquisition strategy.13 Furthermore, while the presence of reputation concerns might induce media outlets to hire “better”journalists/editors (e.g., hire editors with a lower c), these reputations concerns would affect both moderate and non-moderate news media. Therefore, absent other asymmetries among media outlets leading them to choose editors with different skills, the link between ideology and news accuracy pointed out by Proposition 2, is unlikely to disappear.14 At the same time, it would be extremely costly for a media outlet to induce a moderate editor to gather an amount of information such that even extremists citizens would consider this media outlet a valuable source of information.15 Finally, as discussed in section 4, while all citizens ˜e |xe = 1 find the information coming from a moderate editor with preferences x ˆe |xe = 1 < xi < x 2

2

valuable, some of them would find the information coming from an editor with similar idiosyncratic preferences even more valuable. Hence, there will always be a demand for “slanted” news by ideological citizens that a media outlet may capture by simply hiring an ideological editor.16

B.2

Editor’s Influence on Citizens

In the model the utility of the editor depends on her own choice. Nevertheless, even if the editor’s utility were to depend on the citizens’ choice, the information acquisition strategy of the editor would not change. Indeed, the only credible strategy by an editor with idiosyncratic preferences ¯ ∗e . A parallel to the political¯ ∗e upon reaching n xe is to report n∗e upon reaching n∗e and to report n economy literature on citizens-candidates (Osborne and Slivinsky, 1996; Besley and Coate, 1997), might be particularly useful to exemplify the rationale behind this mechanism. In the citizencandidate framework, candidates could in principle gain votes by proposing a platform that is more appealing to a larger number of voters with respect to her own preferred policy. However, this is not a feasible strategy for a candidate simply because voters know that the citizen-candidate could only credible commit to her preferred policy since, once in office, this is the policy that she will pick. Similarly, a citizen-editor is only able to commit to her own stopping thresholds since these are the only ones that could be credible from the viewers’ point of view. That is, the stopping thresholds represent the evidence that an editor requires to be “convinced” about a given 13

Moreover, the cost of acquiring information by editors may be also reinterpreted as a discount factor (see Brocas and Carrillo 2009). In such case, each editor has to decide when to stop gathering information. Hence, by inducing an editor to sample more, a media outlet would also delay the release of the news report which may have a negative effect on the demand for it and, hence, on the profits. 14 Analyzing a model of horizontal and vertical competition among media outlets is beyond the scope of the model. It represents an interesting avenue for future research 15 Indeed, x ˆe → 0 and x ˜e → 1 if and only if n∗e → −∞, n ¯ ∗e → ∞, δ → 1/2 and C → 0. 16 Moreover, it would be cheaper for a media outlet to capture such demand for “slanted” news of nonmoderate citizens by hiring an editor with similar idiosyncratic preferences, rather than hiring a moderate one and provide her with incentives to acquire a large amount of information in both directions.

16

candidate. Any additional evidence that the editor might claim to have found in favor of her “endorsed” candidates will simply not be credible. Indeed, since citizens know the idiosyncratic preferences of the editor, even if she were to try to influence citizens’ choice by over-reporting the number of signals in favor of a given candidate, citizens would still be able to perfectly discount her “bias” and infer the actual stopping threshold (i.e., any n > n ¯ ∗e would be interpreted as n ¯ ∗e and any n < n∗e as n∗e ). Suppose for example that a leftist editor would announce that she would choose an optimal information acquisition rule (stopping threshold) which is “tougher” on the leftist candidate. That is, she announces that she would require more signals in favor of L in order to endorse such candidate. This potentially may lead more individuals to be willing to read her news reports since this would be more informative. However, such announcement would simply not be credible. Indeed, readers cannot monitor the information acquired by editors/journalists. Since the endorsement/report is based on the difference between the number of signals in favor of L with respect to the ones in favor of R (i.e., net evidence), readers know that the editor may simply manipulate the reported number of signals in favor of one candidate or the other. Suppose, for example, that the stopping rule of a leftist editor is such that she would endorse candidate L after having found x more signals in favor of L and, instead, she announces that she will endorse L only after having found y more signals in favor of L (with y > x). Then, as in any model of supply-driven media bias, such an editor may simply selectively omit a subset of the available information once she has reached one of her optimal stopping thresholds. That is, once she arrives to the “x more in favor of L” threshold she may easily show to readers “y more in favor of L” by simply hiding (y − x) signals in favor of R. Therefore, readers would not find credible such announcement by a media editor. Consequently, the media editor is only able to credibly commit to her own optimal information acquisition strategy. Notice that the model could indeed be seen as a special case of a commitment-free mechanism of Bayesian persuasion, as defined by Kamenica and Gentzkow (2011), where the Sender (the editor) can influence the choice of a rational Bayesian Receiver (the citizens) by influencing her beliefs. Specifically, in my setting the fact that the Sender’s preferences depend on the state of the world and acquiring signals is costly, mitigates the incentive compatibility constraints. That is, there is an endogenous commitment mechanism arising from the editor’s idiosyncratic preferences and the cost of drawing a signal. The Receiver knows that the only credible signal realization is the one implicitly defined by the two stopping thresholds of the Sender (i.e., the editor can only credibly commit to such signal acquisition strategy). Any other mechanism would, simply, not be credible. The stopping thresholds represent the net difference in the number of signals in favor of one candidate. Hence, once the editor has reached one of the two thresholds, she has always an incentive to hide signals against the endorsed candidate. Hence, since there is an alignment of preferences between the Sender and the Receiver (i.e., all citizens willing to acquire information from a given editor will have the same ex-post ranking of preferences as the one of the editor), the

17

Sender will truthfully reveal the signal realization. Clearly, in the presence of uncertainty on the editor’s idiosyncratic preferences there would also be uncertainty on the editor’s optimal stopping thresholds. That is, if citizens only knew that   ∗ ∗ B A B xe ∼ g(x) with supp(x) = xA e , xe and xe < xe , then they would also know that ne ∼ g(ne (xe ))   A where nB = n∗ (xB ) < nA = n∗ (xA ), since there is a one-towith supp [g(n∗e (xe ))] = nB e , ne e e e e e e one mapping between preferences and optimal stopping thresholds. Similarly, n ¯ ∗e (xe ) ∼ g(¯ n∗e (xe ))  B A with supp [g(¯ n∗e (xe ))] = n ¯e , n ¯ e where n ¯B ¯ ∗e (xB ¯A ¯ ∗e (xA e = n e ) < n e = n e ). In presence of such additional source of uncertainty, the editor will have an incentive to over-report signals in favor of the preferred candidate once she has reached one of the two stopping thresholds. That is, such uncertainty would introduce in the model a “supply-driven” bias in news reports since the editor would have an incentive to bias its news reports by selectively omitting a subset of her information. Nevertheless, if the editor had to report n ¯A e , citizens’ posterior beliefs would be 17 That is, citizens will still be able to infer the interval in which the µ(¯ nA n∗e (xe )|¯ nA e ) = µ(E(¯ e )).

optimal editor’s stopping threshold lies and discount their posterior beliefs accordingly. Hence, the main mechanism and intuition of the model would not change. Obviously, the more ideologically distant from the endorsed candidate the editor is believed to be, the more influential her reports will be. In other words, the editor’s endorsement will be stronger: i) the more moderate the editor is believed to be, upon endorsing the ideologically closer candidate; ii) the less moderate the editor is believed to be, upon endorsing the ideologically least preferred candidate. Hence, in most of the cases (i.e., when endorsing the ideologically closer candidate), an editor would like to be believed to be as “unbiased” (i.e., moderate) as possible. Indeed, consistent with the theoretical predictions of the model, the empirical analysis of Chiang and Knight (2011) shows that the degree of influence of a newspaper on voters depends on the “credibility” of the endorsement.

B.3

Multiple Sources of Information

Throughout the analysis, it was assumed that citizens watch at most one media outlet. Nevertheless, while such assumption greatly simplifies the analysis, the intuition and the main results of the model do not rely on it. Indeed, if citizens were to acquire information from multiple sources, the incentives of media outlets to choose ideological editors would only be reinforced. For any citizen, watching two media outlets with a moderate editor has the same value of watching only one. Specifically, after having observed the news report of a moderate editor, watching an additional media outlet with another moderate editor would either not change the citizen’s ranking of preferences, or it would lead citizen’s posterior beliefs to be equal to the prior (i.e., the two reports would just “cancel” each other). Hence, if citizens could access multiple sources of information, the incentives of media outlets to differentiate their products by hiring ideological editors would, indeed, be higher. 17

Similarly, upon reporting n ¯B nB n∗e (xe )|¯ nB e , citizens’ posterior beliefs would be µ(¯ e ) = µ(E(¯ e )).

18

Appendix C The following graph illustrates the relationship between the perceived quality and fairness of US news organizations from 1985 to 2011 (Pew Research Center, 2012). In particular, the perceived quality of news organizations, in a given year, is measured in terms of the percentage of individuals who think that news organizations get their facts straight. The perceived fairness of news organizations is measured as the percentage of individuals who think that news media deal fairly

60 50

1985

1992

40

1988 2007 1997 2003 2001

30

2005

2009 2011

20

News organizations get their fact straight (%)

with all sides (on political and social issues).

15

20

25

30

35

News organizations deal fairly with all sides (%) Fitted values

Year

Fig. C1. Perceived quality and fairness of US news organizations. (Source: Pew Research Center, 2011)

The following graph illustrates the evolution of perceived quality and fairness of US news organizations over time along with the trend in the percentage of individuals considering themselves as

10

20

30

40

50

politically independent.

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 Year News organizations deal fairly with all sides (%) News organizations get their fact straight (%) Percentage of individuals who consider themselves as independents

Fig. C2. Evolution of perceived quality and fairness of news organization (US, 1988-2011) (Source: Pew Research Center, 2011)

19

As the graph shows, the negative trends in accuracy and “fairness” of US news organizations do not seem to have been paralleled by a similar negative trend in the percentage of individuals who consider themselves as politically independent. Hence, there is no evidence that the negative correlation between accuracy and fairness of US news organizations (in a given year) is driven by an upward trend in the polarization of the US electorate. Indeed, as further emphasized by Figure C3, there is no statistically significant correlation between the percentage of individuals who consider themselves as politically independent and the percentage of individuals who consider news organizations accurate (i.e., affirmative answers to the question “News organizations get their

60 50

1992

40

1988

2007 1997

30

2001

20052003

2009 2011

20

News organizations get their fact straight (%)

facts straight”), in a given year.

30

32

34

36

38

Individuals who consider themselves as politically independent(%) Fitted values

News organizations get their fact straight (%)

Fig. C3. Perceived quality of news organization and percentage of individuals who consider themselves as politically independent (US, 1988-2011) (Source: Pew Research Center, 2011)

Similarly, Figure C4 shows the absence of any statistically significant correlation between the percentage of individuals who consider themselves as politically independent and the percentage of individuals who consider news organizations being unbiased (i.e., affirmative answers to the question “News organizations deal fairly with all sides”).

20

News organizations deal fairly with all sides (%) 35 15 20 25 30

1992 1988

1997 2001

2003

2007

2005

2009 2011

30 32 34 36 38 Individuals who consider themselves as politically independent (%) Fitted values

News organizations deal fairly with all sides (%)

Fig. C4. Perceived fairness of news organizations and percentage of individuals who consider themselves as politically independent (US, 1988-2011) (Source: Pew Research Center, 2011)

Overall, the above graphs present evidence consistent with the theoretical prediction of Proposition 2. That is, in the presence of more moderate news media (i.e., in the language of the model: news media having editors with a “fair and balanced” information acquisition strategy), news media are perceived to be more accurate. At the same time, the evidence drawn from the Pew Research Center (2011) data shows that the negative correlation between slant and accuracy does not seem to have any relationship with a negative trend in the percentage of independent voters (i.e., with a polarization of the US electorate).

21

Appendix D D.2

Probabilities of reaching a stopping threshold

This section presents a complete formal derivation of the probabilities of hitting the two stopping thresholds. These probabilities, and their characterization, are analogous to the ones present in Lemma 1 of Brocas and Carrillo (2007). ¯ ∗e (and not n∗e ) when the lower stopping threshLet q s (µ(¯ n∗e ), µ(n∗e )) be the probability of reaching n ¯ ∗e and the state is s. Clearly, 1 − q s (µ(¯ n∗e ), µ(n∗e )) old is n∗e and the upper stopping threshold is n represents the probability of reaching threshold n∗e when the state is s. Let, instead, q(µ(¯ n∗e ), µ(n∗e )) be the unconditional probability of reaching n ¯ ∗e . Moreover, denote η s (n) the probability of reaching a difference of signals n ¯ ∗e before reaching a difference of signals n∗e when the current difference of signals is n and the true state is s. Then, it is clearly the case that: η s (n∗e ) ≡ 0; η s (¯ n∗e ) ≡ 1 and η s (0) = q s (µ(¯ n∗e ), µ(n∗e )) Moreover: η r (n) = θ · η r (n + 1) + (1 − θ) · η r (n − 1),

∀n ∈ {n∗e + 1, ....., n ¯ ∗e − 1}

(D-1)

η l (n) = (1 − θ) · η l (n + 1) + θ · η l (n − 1),

∀n ∈ {n∗e + 1, ....., n ¯ ∗e − 1}

(D-2)

from (D-1), we have: 1 (1 − θ) r η r (n + 1) − η r (n) + η (n − 1) = 0 θ θ The generic solution to this second-order difference equation is of the form: η r (n) = κ1 · z1n + κ2 · z2n where (κ1 , κ2 ) are constants and (z1 , z2 ) are the roots of the second order equation: 1 (1 − θ) x2 − x + =0 θ θ Thus: z1 =

(1 − θ) and z2 = 1 θ

In order to determine (κ1 , κ2 ) it is possible to use the fact that η r (n∗e ) = 0 and η r (¯ n∗e ) = 1. That is: η

r

(n∗e )

η

r

(¯ n∗e )

 = 0 =⇒ κ1

1−θ θ

n∗e

1−θ θ

n¯ ∗e

and  = 1 =⇒ κ1

22

+ κ2 = 0

+ κ2 = 1

1−θ θ ,

Denoting, as in Proposition 1, λ =

the above equations imply: ∗

κ1 =

λn¯ ∗e

1 λne and κ = 2 ∗ ∗ − λne λne − λn¯ ∗e

Thus, the general solution is: ∗

η r (n) =

1 − λn−ne ∗ ∗, 1 − λn¯ e −ne

∀n ∈ {n∗e , ....., n ¯ ∗e }

(D-3)

note that from (D-1) and (D-2), the case s = l is obtained by simply switching θ and 1 − θ. That is:



η l (n) = Then given that µ(¯ n∗e ) =

1 ¯∗ e 1+λn





λn¯ e −n − λn¯ e −ne , ∗ ∗ 1 − λn¯ e −ne

and µ(n∗e ) = ∗

λn¯ e =

1 ∗ 1+λne

¯ ∗e } ∀n ∈ {n∗e , ....., n

(D-4)

:

1 − µ(¯ n∗e ) 1 − µ(n∗e ) n∗e = and λ µ(¯ n∗e ) µ(n∗e )

(D-5)

hence combining (D-3),(D-4) and (D-5): 1 − 2µ(n∗e ) µ(¯ n∗e ) µ(¯ n∗e ) − µ(n∗e ) 1 − 2µ(n∗e ) η l (0) = q l (µ(¯ n∗e ), µ(n∗e )) = (1 − µ(¯ n∗e )) µ(¯ n∗e ) − µ(n∗e )

η r (0) = q r (µ(¯ n∗e ), µ(n∗e )) =

Hence:

D.3

1 1 1 − 2µ(n∗e ) q(µ(¯ n∗e ), µ(n∗e )) = η r (0) + η l (0) = 2 2 2 (µ(¯ n∗e ) − µ(n∗e ))

Marginal Viewers

The marginal viewer represents the viewer who is indifferent between watching and not watching the media outlet’s reports. Specifically, there will be two marginal viewers. One representing the most rightist citizen willing to watch news reports from a media outlet having an editor with idiosyncratic preferences xe . The other one representing the most leftist citizen willing to watch such news reports. That is, there will be a x ˆe = x ˆe (xe ) and a x ˜e = x ˜e (xe ) with x ˆe < x ˜e such that only citizens with xi ∈ [ˆ xe , x ˜e ] will watch the news reports.18  Let’s start analyzing the marginal viewer for xi < 21 . Then Ui (N W ) = Ui L| 12 and since by (13) n∗e < 0 < n ¯ ∗e , it must be the case that: Ui (L|µ(n∗e )) > Ui (R|µ(n∗e )) 18

Notice that it could also be the case that x ˆe >

1 2

or x ˜e <

23

1 2

but, clearly, not both.

Moreover, the following individual rationality constraint must be satisfied for leftist citizens: Ui (L|µ(¯ n∗e )) < Ui (R|µ(¯ n∗e ))

(IRL )

otherwise, if Ui (L|µ(¯ n∗e )) > Ui (R|µ(¯ n∗e )) (i.e., if citizen i would always prefer alternative L regardless of watching or not the news reports) then watching the news reports would never be ex-post rational given the cost C. Thus the marginal leftist viewer will be the one having idiosyncratic preferences x ˆe such that:   1 1 − 2µ(n∗e ) 2µ(¯ n∗e ) − 1 ∗ )) + = U (L|µ(n Ui (R|µ(¯ n∗e )) − C Ui L i e 2 2 [µ(¯ n∗e ) − µ(n∗e )] 2 [µ(¯ n∗e ) − µ(n∗e )] that is: x ˆe =

C 1 − δ(2µ(¯ n∗e ) − 1) + 2 2 Pr(n = n ¯ ∗e )

Notice also that the ex-post rationality constraint (IRL ) is satisfied as long as xi >

(D-6) 1 2

− δ(2µ(¯ n∗e ) −

1) = xmin . Hence, since x ˆe > xmin , such constraint is automatically satisfied for any citizen willing to watch the news reports.  Let’s now focus on the marginal viewer for xi > 21 . Then Ui (N W ) = Ui R| 12 and since by (13) n∗e < 0 < n ¯ ∗e , it must be the case that: Ui (R|µ(¯ n∗e )) > Ui (L|µ(¯ n∗e )) Moreover, the following individual rationality constraint must be satisfied for rightist citizens: Ui (L|µ(n∗e )) > Ui (R|µ(n∗e ))

(IRR )

otherwise, if Ui (L|µ(n∗e )) < Ui (R|µ(n∗e )) (i.e., if citizen i would always prefer alternative R regardless of watching or not the news reports) then watching the news reports would not be ex-post rational given the cost C. Thus the marginal rightist viewer will be the one having idiosyncratic preferences x ˜e such that:   1 2µ(¯ n∗e ) − 1 1 − 2µ(n∗e ) ∗ Ui R = U (L|µ(n )) + Ui (R|µ(¯ n∗e )) − C i e 2 2 [µ(¯ n∗e ) − µ(n∗e )] 2 [µ(¯ n∗e ) − µ(n∗e )] that is: x ˜e =

1 C + δ(1 − 2µ(n∗e )) − 2 2 Pr(n = n∗e )

Notice also that the ex-post rationality constraint (IRR ) is satisfied as long as xi <

(D-7) 1 2

+ δ(1 −

2µ(n∗e )) = xmax . Hence, since x ˜e < xmax , such constraint is automatically satisfied for any citizen willing to watch the news reports

24

Citizen-Editors' Endogenous Information Acquisition ...

Email: [email protected] ...... Notice that for any non-uniform distribution satisfying Condition A the mass of citizens is strictly ..... Notice that the model could indeed be seen as a special case of a commitment-free mechanism ... preferences between the Sender and the Receiver (i.e., all citizens willing to acquire ...

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